scholarly journals Gaussian curvature of the Bergman metric with weighted Bergman kernel on the unit disc

2012 ◽  
Vol 45 (3) ◽  
Author(s):  
Marzena Szajewska
Keyword(s):  
Weight Function ◽  
Bergman Kernel ◽  
Gaussian Curvature ◽  
Unit Disc ◽  
Bergman Metric ◽  
Weighted Bergman Kernel

AbstractIn the paper Gaussian curvature of Bergman metric on the unit disc and the dependence of this curvature on the weight function has been studied.

Bergman kernel on Riemann surfaces and Kähler metric on symmetric products

2019 ◽  
Vol 30 (14) ◽  
pp. 1950071
Author(s):  
Anilatmaja Aryasomayajula ◽  
Indranil Biswas
Keyword(s):  
Bergman Kernel ◽  
Riemann Surfaces ◽  
Symmetric Product ◽  
Volume Form ◽  
Bergman Metric ◽  
Poincaré Metric ◽  
Kähler Metric ◽  
Fixed Dimension ◽  
Holomorphic Sections ◽  
Kahler Metric

Let [Formula: see text] be a compact hyperbolic Riemann surface equipped with the Poincaré metric. For any integer [Formula: see text], we investigate the Bergman kernel associated to the holomorphic Hermitian line bundle [Formula: see text], where [Formula: see text] is the holomorphic cotangent bundle of [Formula: see text]. Our first main result estimates the corresponding Bergman metric on [Formula: see text] in terms of the Poincaré metric. We then consider a certain natural embedding of the symmetric product of [Formula: see text] into a Grassmannian parametrizing subspaces of fixed dimension of the space of all global holomorphic sections of [Formula: see text]. The Fubini–Study metric on the Grassmannian restricts to a Kähler metric on the symmetric product of [Formula: see text]. The volume form for this restricted metric on the symmetric product is estimated in terms of the Bergman kernel of [Formula: see text] and the volume form for the orbifold Kähler form on the symmetric product given by the Poincaré metric on [Formula: see text].

scholarly journals On weights which admit the reproducing kernel of Bergman type

1992 ◽  
Vol 15 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Zbigniew Pasternak-Winiarski
Keyword(s):  
Unit Disk ◽  
Bergman Kernel ◽  
Reproducing Kernel ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Admissible Weight ◽  
Necessary And Sufficient ◽  
Weighted Bergman Kernel

In this paper we consider (1) the weights of integration for which the reproducing kernel of the Bergman type can be defined, i.e., the admissible weights, and (2) the kernels defined by such weights. It is verified that the weighted Bergman kernel has the analogous properties as the classical one. We prove several sufficient conditions and necessary and sufficient conditions for a weight to be an admissible weight. We give also an example of a weight which is not of this class. As a positive example we consider the weightμ(z)=(Imz)2defined on the unit disk inℂ.

Weighted Sub-Bergman Hilbert spaces in the unit ball of ℂn

2020 ◽  
Vol 7 (1) ◽  
pp. 124-132
Author(s):  
Renata Rososzczuk ◽  
Frédéric Symesak
Keyword(s):  
Unit Ball ◽  
Hilbert Spaces ◽  
Bergman Kernel ◽  
Holomorphic Functions ◽  
Sufficient Condition ◽  
Defect Operators ◽  
Weighted Bergman Kernel

AbstractIn this note, we study defect operators in the case of holomorphic functions of the unit ball of ℂn. These operators are built from weighted Bergman kernel with a holomorphic vector. We obtain a description of sub-Hilbert spaces and we give a sufficient condition so that theses spaces are the same.

scholarly journals A note on the harmonic quasiconformal diffeomorphisms of the unit disc

Filomat ◽  
2015 ◽  
Vol 29 (2) ◽  
pp. 335-341
Author(s):  
Miljan Knezevic
Keyword(s):  
Gaussian Curvature ◽  
Unit Disc ◽  
Quasiconformal Mappings ◽  
Conformal Metrics

We analyze the properties of harmonic quasiconformal mappings and by comparing some suitably chosen conformal metrics defined in the unit disc we obtain some geometrically motivated inequalities for those mappings (see for instance [15, 17, 20]). In particular, we obtain the answers to many questions concerning these classes of functions which are related to the determination of different properties that are of essential importance for validity of the results such as those that generalize famous inequalities of the Schwarz-Pick type. The approach used is geometrical in nature, via analyzing the properties of the Gaussian curvature of the conformal metrics we are dealing with. As a consequence of this approach we give a note to the co-Lipschicity of harmonic quasiconformal self mappings of the unit disc at the origin.

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